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A Comment on “Subsidization of Urban Public Transport and the Mohring Effect” by Ian Savage Northwestern University and Kenneth A. Small University of California Irvine Correspondence Address Professor Ian Savage Department of Economics Northwestern University 2001 Sheridan Road Evanston, IL 60208 Ph: +1-847-491-8241 Fax: +1-847-491-7001 e-mail: ipsavage@northwestern.edu April 22, 2009 Abstract Van Reeven (2008) argues that the Mohring effect is not relevant to the determination of transit subsidies because a profit-maximizing monopolist would supply frequencies that are the same as, or greater than, those that are socially optimal. We find that his results depend on the reduction or elimination of the effect of fares on demand, causing optimal prices to be indeterminate within broad ranges. Consequently, his model is an unsatisfactory tool for discussing subsidies in general, and the optimal combination of fare and frequency in particular. 11.0 Introduction In a classic 1972 paper, Herbert Mohring argued that transit subsidies could be justified because of the scale economies conferred on riders. Subsidies increase ridership, the ridership increase engenders higher service frequencies, and the higher frequencies reduce the average waiting times at stops. A recent paper by Peran van Reeven (2008) attempts to refute this line of argument by advancing a proposition that a profit-maximizing monopolist will ...
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1
A Comment on “Subsidization of Urban Public Transport and the Mohring Effect”
by
Ian Savage
Northwestern University
and
Kenneth A. Small
University of California Irvine
Correspondence Address
Professor Ian Savage
Department of Economics
Northwestern University
2001 Sheridan Road
Evanston, IL 60208
Ph:
+1-847-491-8241
Fax: +1-847-491-7001
e-mail: ipsavage@northwestern.edu
April 22, 2009
Abstract
Van Reeven (2008) argues that the Mohring effect is not relevant to the determination of transit
subsidies because a profit-maximizing monopolist would supply frequencies that are the same as,
or greater than, those that are socially optimal.
We find that his results depend on the reduction
or elimination of the effect of fares on demand, causing optimal prices to be indeterminate within
broad ranges.
Consequently, his model is an unsatisfactory tool for discussing subsidies in
general, and the optimal combination of fare and frequency in particular.
2
1.0 Introduction
In a classic 1972 paper, Herbert Mohring argued that transit subsidies could be justified because
of the scale economies conferred on riders.
Subsidies increase ridership, the ridership increase
engenders higher service frequencies, and the higher frequencies reduce the average waiting
times at stops.
A recent paper by Peran van Reeven (2008) attempts to refute this line of
argument by advancing a proposition that a profit-maximizing monopolist will choose to provide
a frequency of service that, depending on the particular parameters within a tightly defined
model, is the same as or greater than the frequency that maximizes social welfare.
From this,
van Reeven concludes that “economies of scale do not constitute a justification for general
subsidisation of urban public transport” (abstract).
1
But do his results really undermine the case
for public transport subsidies arising from the “Mohring effect”?
A major limitation of van Reeven’s model is that price has no effect on welfare within a
range of conditions, because within certain ranges transit demand is perfectly price inelastic,
making it impossible to discuss optimal pricing. Mohring’s original paper had a similar
assumption, but this was a harmless assumption made for convenience because he considered
only cases where welfare optimization implies marginal-cost pricing. Furthermore, the extensive
theoretical and empirical literature that is derived from Mohring’s work has not been so
restrictive.
2.0 A Model with Price Sensitivity
2.1 Consumers who do not know the timetable
In Section 3.1 of van Reeven’s paper, consumers who arrive randomly at a transit stop without
knowledge of a timetable all have the same reservation price — call it
p
r
— which depends on
1
In the paper’s concluding section, van Reeven more properly qualifies this statement: “… do
not
necessarily
constitute a justification for subsidising urban public transport” (page 357,
emphasis added).
3
frequency; the demand curve therefore consists of two vertical line segments joined by a
horizontal section at price
p
r
.
The easiest way to introduce demand elasticity within van Reeven’s framework is to
allow for a continuous variation in reservation prices.
This is the approach taken by Frankena
(1983) and Jansson (1993) in their reworking of the Mohring model.
Using van Reeven’s
terminology, and his assumption that service frequency is provided at a fixed unit cost (
c
), the
monopolist profit (
π
) will be given by:
(1)
cf
f
p
G
pD
−
=
)]
,
(
[
π
where
p
is the fare,
f
is service frequency (departures per hour) and
D
(
⋅
) is the demand function.
Demand depends on the generalized price of travel,
G
(
⋅
), which is a function of fare and average
waiting time, the latter depending inversely on frequency so that
∂
G
/
∂
f
<0.
Profit maximization
produces the following first order conditions:
(2)
0
)]
,
(
[
)]
,
(
[
=
+
∂
∂
=
∂
∂
f
p
G
D
P
f
p
G
D
p
p
π
and
(3)
0
)]
,
(
[
=
−
∂
∂
=
∂
∂
c
f
f
p
G
D
p
f
π
Social welfare (
W
) is defined as the combination of area under the demand curve and
above the equilibrium level of generalized price plus the profit/loss of the operator:
(4)
∫
∞
−
+
∂
=
)
,
(
)]
,
(
[
]
[
f
p
G
cf
f
p
G
pD
x
x
D
W
Welfare maximization produces the first order conditions;
4
(5)
(
)
[
]
(
)
[
]
0
,
,
)
,
(
)]
,
(
[
=
∂
∂
+
+
∂
∂
−
=
∂
∂
p
f
p
G
D
p
f
p
G
D
p
f
p
G
f
p
G
D
p
W
and
(6)
(
)
[
]
0
,
)
,
(
)]
,
(
[
=
−
∂
∂
+
∂
∂
−
=
∂
∂
c
f
f
p
G
D
p
f
f
p
G
f
p
G
D
f
W
In setting fares, equation (2) indicates that the profit-maximizing monopolist (assuming
that it decides to produce) will set a positive price consistent with marginal revenue equaling
zero, whereas equation (5) simplifies to indicate that the socially-optimal fare is zero.
Not
surprisingly, we can conclude that the monopolist will charge a higher than optimal fare.
In setting frequency, equation (3) indicates that the profit-maximizing monopolist will
equate the (constant) marginal cost of providing additional frequency to the marginal revenue
from that frequency.
However, from equation (6), welfare will be maximized when the same
marginal cost is equated to the same marginal revenue
augmented by
the Mohring benefits of
reduced waiting times, –
D
∂
G
/
∂
f
(which are positive because
∂
G
(
⋅
)
/
∂
f
<0).
Therefore,
for a given
fare,
the profit-maximizing monopolist will provide a lower frequency than is socially optimal.
Furthermore, the monopolist charges too high a fare, leading one to suspect that its frequency is
even further from the first-best optimal one, although this need not always be true.
2
Therefore, determination of whether a profit-maximizing monopolist produces a larger or
smaller frequency than is socially optimal will depend on the parameters and functional forms
governing how marginal revenues and marginal Mohring benefits depend on frequency.
While
we cannot be sure which situation will produce a higher frequency, we can reject van Reeven’s
conclusion from proposition 1 that “
any
subsidy has an adverse effect” (page 355, emphasis
added).
Properly structured subsidies to a profit-maximizing monopolist will enable fares to be
reduced toward their socially optimal level (zero in this formulation), and possibly lead to higher
2
Because the social welfare-maximizing fare is zero, equation (6) simplifies to tell us that this
first-best optimal frequency occurs when the marginal cost of additional frequency equals the
marginal Mohring benefits.
5
frequencies, if that is desirable.
Frankena (1983) provides a very extensive discussion of fare
and frequency choice in these circumstances.
We note in passing that van Reeven’s model removes much of the richness of Mohring’s
model.
Additional riders may impose negative as well as positive externalities on other riders, as
buses may have to stop more often to pick up and drop off passengers, and have longer dwell
times at existing stops.
Moreover, the slower average vehicle speeds have cost implications for
the bus company as more vehicles are required to provide a set number of departures per hour.
2.2 Consumers who know the timetable
The general model just formulated also describes the case when passengers know the timetable,
and plan their arrival time at transit stops.
At a more detailed level, models of this case are
considerably more complex than those used to describe random arrivals at stops.
Panzar (1979),
for airlines, and Jansson (1993), for transit, provide extensive analyses.
Transit models in which
consumers decide whether or not to travel depending on the difference between their desired
departure time and the known schedule are found to be analogous to the standard industrial
organization model of quality choice (with frequency as the measure of quality).
It is a standard result in this literature that a profit-maximizing monopolist may produce a
higher or lower level of quality than is socially optimal (Spence, 1975; Sheshinski, 1976).
This
is because the profit-maximizing monopolist sets quality based on the preferences of the
marginal consumer, while the social-welfare maximizing firm bases it on the preferences of the
average consumer.
Because the profit-maximizing monopolist also raises price and restricts the
number of passengers, we cannot be sure of the relative magnitude of the preferences of the
monopolist’s marginal consumer and the average consumer who would purchase in a social-
welfare maximizing world.
Frankena (1983) discusses the Spence/Sheshinski result in a transit
context.
Van Reeven’s consumers who follow a timetable do have some elasticity (within a
certain range of parameters), because they do not all have the same preferred schedule.
Nevertheless, the model lacks the usual features that create a downward sloping demand curve,
namely either price-responsiveness in the amount of travel by each individual or heterogeneity in
individuals’ reservation prices for making a fixed number of transit trips.
This is why van
6
Reeven obtains a non-ambiguous result comparing monopoly with optimal frequency, leading
him to advocate a tax on frequency; and it is why once again the optimal fare is undefined, thus
obscuring the result following from (5) in our formulation.
2.3 Summary
By oversimplifying the demand structure, van Reeven’s model is rendered incapable of saying
much about optimal prices.
This makes it an unsatisfactory tool for discussing subsidies or the
optimal combination of fare and frequency.
3.0 Empirical Evidence
Because it is theoretically indeterminate whether a profit-maximizing monopolist will produce
too much or too little frequency, the issue of the role of the Mohring effect in determining
optimal subsidies becomes an empirical one.
The empirical literature has not directly addressed
the issue raised by van Reeven (monopolist’s profit-maximizing choice versus social optimality),
but rather has analyzed whether providing additional subsidies to encourage operators to lower
their
existing
fare and/or expand their
existing
frequencies is socially desirable (Dodgson, 1987;
Glaister, 1987, 2001; Savage and Schupp, 1997; Small and Gómez-Ibáñez, 1999; Savage, 2008;
Parry and Small, 2009).
Some of these analyses have found situations where subsidies to expand
frequencies are socially worthwhile, and the majority of the benefits accrue from the Mohring
effect.
Another interesting finding from the empirical work is that, for a given budget constraint,
transit agencies typically have a suboptimal mix of fare and frequency.
Too high a frequency is
produced at too high a fare.
Thus, even in an optimum constrained by a fixed subsidy budget
(perhaps zero), the scale economies inherent in the Mohring effect are likely to substantially
influence the desired operating configuration.
7
4.0 Other Issues
Several authors have noted the institutional problem that subsidies are often used to cover an
increase in unit costs rather than to improve services by lowering fares and/or increasing
frequencies.
There is a debate as to whether the rapid increase in subsidies in the 1970s was the
cause of, or a consequence of, rising unit costs (Pickrell, 1985).
We agree with van Reeven that
this is an important practical consideration in considering transit subsidies, especially the
structure they should take.
An even more interesting question, mentioned in passing by van Reeven, is that there are
many goods produced under increasing returns to scale.
Indeed, one can argue that this is typical
of large segments of an economy, as recognized in models like that of Dixit and Stiglitz (1977).
Why not subsidize all of these products?
To examine this question carefully would require a
general-equilibrium model with a tax system that creates distortions by causing one or more
prices (
e.g.
of labor) to differ between suppliers and demanders.
While it is difficult to make
generalizations, several authors have examined transport within such a model and found that
optimal prices still include a Mohring effect term, pretty much in the form described by the
simple model here, but modified by other terms that capture the degree of existing tax distortions
(perhaps summarized in a marginal cost of public funds) and the degree of substitutability
between public transport and labor supply.
3
5.0 Concluding Comments
Even excluding the problems caused by the suppression of the role of price, van Reeven
overstates his case within his own model.
Under many, perhaps most, parameters his model
produces the result that a private monopolist produces unsatisfactory results.
For example, in his
Proposition 2 there is a range of reservation prices covering a factor of 1.41 for which the
monopolist fails to produce at all, even though an optimal operator would do so.
This range
consists of the entire range of demand conditions between that where optimal service is zero to
that for which the monopolist’s service frequency is so high that the transit service meets
3
See, for example, Parry and Bento (2001), van Dender (2003).
8
everyone’s reservation price.
In van Reeven’s Proposition 1, the monopolist offers the optimal
frequency but charges the highest price compatible with achieving non-zero demand.
Surely
then it is an exaggeration to say that “the results in this paper actually favour private operation”
(page 358).
References
Dixit, A. and J. Stiglitz (1977): ‘Monopolistic Competition and Optimum Product Diversity’,
American Economic Review
, 67, 297-308.
Dodgson, J.S. (1987): ‘Benefits of Changes in Urban Public Transport Subsidies in the Major
Australian Cities’, in S. Glaister (ed.)
Transport Subsidy
, Policy Journals, Newbury.
Frankena, M.W. (1983): ‘The Efficiency of Public Transport Objectives and Subsidy Formulas’,
Journal of Transport Economics and Policy
, 17, 67-76.
Glaister, S. (1987): ‘Allocation of Urban Public Transport Subsidy’, in S. Glaister (ed.)
Transport Subsidy
, Policy Journals, Newbury.
Glaister, S. (2001): ‘The Economic Assessment of Local Transport Subsidies in Large Cities’, in
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Any More Fares? Delivering Better Bus Services
, Institute for Public
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Journal of
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, 27, 33-50.
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American
Economic Review
, 69, 92-95.
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Pricing’,
Scandinavian Journal of Economics
, 103, 645–71.
Parry, I.W.H. and K.A. Small (2009): ‘Should Urban Transit Subsidies be Reduced?’
American
Economic Review
, 99(3).
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Journal of Transport Economics and Policy
, 19, 281-298.
9
Savage, I. (2008): ‘The Dynamics of Fare and Frequency Choice in Urban Transit’, Mimeo,
Northwestern University.
Savage, I. and A. Schupp (1997): ‘Evaluating Transit Subsidies in Chicago’,
Journal of Public
Transportation
, 1, 93-117.
Sheshinski, E. (1976): ‘Price, Quality and Quantity Regulation in Monopoly Situations’,
Economica
, 43, 127-137.
Small, K.A. and J.A. Gómez-Ibáñez (1999): ‘Urban Transportation’, in Cheshire, P. and E.S.
Mills (eds.)
Handbook of Regional and Urban Economics, Volume 3: Applied Urban
Economics
, North-Holland, Amsterdam.
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, 6, 417-
429.
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Economics
, 105, 295-310.
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Journal of Transport Economics and Policy,
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